Minimum Area of a Convex Grid Polygon
A symmetrical convex grid polygon is a polygon such that:
- All its vertices have integer coordinates.
- All its internal angles are strictly smaller than $180^\circ$.
- It has both horizontal and vertical symmetry.
For example, the left polygon is a convex grid polygon which has neither horizontal nor vertical symmetry, while the right one is a valid symmetrical convex grid polygon with six vertices:
Define $A(N)$, the minimum area of a symmetrical convex grid polygon with $N$ vertices.
You are given $A(4) = 1$, $A(8) = 7$, $A(40) = 1039$ and $A(100) = 17473$.
Find $A(1000)$.